Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

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Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

Lyonell Boulton, Nabile Boussaid

To cite this version:

Lyonell Boulton, Nabile Boussaid. Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials. 2008. �hal-00308843�

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OF THE EIGENSTATES OF DIRAC OPERATORS WITH RADIALLY SYMMETRIC POTENTIALS

LYONELL BOULTON AND NABILE BOUSSAID

Abstract. We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments.

1. Introduction

The free Dirac operator acting on 4-spinors ofL²(R³)⁴ is determined by the first order differential expression

D:=α·P +β =−i

3

X

k=1

αk∂k+β, where α= (α1, α2, α3), and the Pauli-Dirac matrices are:

αi =

0 σ_i σ_i 0

and β=

IC² 0 0 −IC²

, for σ1 =

0 1 1 0

, σ2 =

0 −i i 0

and σ3 =

1 0 0 −1

. We assume that the units are fixed so that m =c= ~= 1. Standard arguments involving the Fourier transform show that D defines a self- adjoint operator with domain H¹(R³)⁴ and that the spectrum of D is

Spec(D) = (−∞,−1]∪[1,∞).

Spherically symmetric potentials are Hermitean 4×4 matrix multi- plication operators, V, acting on L²(R²)⁴, such that C0^∞(R³ \ {0}) ⊂

Date: July 2008.

Key words and phrases. Dirac operators, quadratic projection methods, spectral pollution.

1

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Dom(V) and

e^iϕn^·^SV(R⁻¹x)e⁻^iϕn^·^S =V(x), ∀x∈R³, ∀ϕ∈[0,4π), where

S = 1 2

0 σ σ 0

is the spin operator, and Ris the matrix of the rotation of angleϕ and axis n. Here Dom(V) denotes the maximal domain of V.

Spherically symmetric potentials may be constructed from maps φsc,el,am :R−→Rvia

(1.1) V(x) =φsc(|x|)β+φel(|x|)I^C⁴ +iφam(|x|)βα· x

|x|.

The subscripts “sc”, “el” and “am”, stand for “scalar”, “electric” and

“magnetic” potential, respectively. Radial symmetry on the electric potential, for instance, is a consequence of the assumption that the atomic nucleus is pointwise and the electric forces are isotropic in an isotropic medium like the vacuum. In the particular caseφsc=φam = 0 and φel(r) =γ/r, |γ|<√

3/2, H describes the motion of a relativistic electron in the field created by an atomic nucleus.

If V is subject to suitable smallness and regularity conditions (such as those ensuring thatV is relatively compact with respect toD), then H :=D+V defines an essentially self-adjoint operator inC₀^∞(R³\{0})⁴ with self-adjoint extension domain H¹(R³)⁴ and

(1.2) Spec_ess(H) = Spec(D) = (−∞,−1]∪[1,∞),

see [Tha92, Theorems 4.2 and 4.16]. In fact there are known conditions, satisfied by potentials of practical interest (see e.g. [BG87, Theorems 6 and A]), preventing the existence of embedded eigenvalues.

The addition of a non-zero potential might give rise to a non-empty discrete spectrum in the gap (−1,1). These eigenvalues can only be found explicitly for a few simple systems which do not go much be- yond the case of a single electron embedded in a field created by an atomic nucleus, see (4.1). For more complicated potentials, one has to rely either on asymptotic techniques (cf. [BL94], [GL99], [Sch03] and references therein) or on numerical estimations.

Few robust computational procedures are currently available to estimate numerically the eigenvalues ofH, see [Dya90], [DES00], [DESV00], [DES03], [DEL07], [LLT02] and references therein. Since H is strongly indefinite, a direct implementation of the projection method is not pos- sible due to variational collapse. Under no further restriction on the potential or the reduction basis, accumulation points of eigenvalues of the finite dimensional approximate operator which do not belong to

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Spec(H) might appear, see [SH84] and [Dya90]. This phenomenon is known as spectral pollution.

The aim of the present paper is to investigate the applicability of the so called quadratic projection method for finding eigenvalues and eigenfunctions of H. This method has been recently studied in an abstract setting (see [Sha00], [LS04], [Bou06] and [Bou07]) and it has already been applied with some success to crystalline Schr¨odinger operators, [BL07], and magnetohydrodynamics, [Str08]. In this approach, explained at length in Section 3, the underlying discretised eigenvalue problem is quadratic in the spectral parameter (rather than linear), and has non-real eigenvalues. Its main advantage over a standard projection method is its robustness: it never pollutes and it always provides a posteriori two-sided estimates of the error of computed eigenvalues.

Section 3.1 is devoted to a self-contained description of the quadratic projection method. In Theorem 1 we present an alternative proof of Shargorodsky’s non-pollution Theorem [LS04, Theorem 2.6]. Addition- ally we show that information about eigenfunctions can be recovered from the quadratic projection method, see (3.4).

In Section 4 we test the practical applicability of the numerical scheme proposed in Section 3 by reporting on various numerical experiments. As benchmark potentials we have chosen the purely coulombic, sub-coulombic and inverse harmonic electric potentials. In order to perform these numerical experiments, we split the space into upper and lower spinor component, after writing the problem in radial form. Spectral pollution in the standard projection method using this decomposition and the consequences of unbalancing the number of upper/lower components is discussed in Section 2.

We have chosen a basis of Hermite functions to reduce the continu- ous problem into finite-dimensional from. In Section 3.3 we perform a rigourous convergence analysis of the quadratic method using this basis.

This analysis relies upon the general result [Bou07, Theorem 2.1]. The surprising numerical evidence included in Section 4.4 strongly suggest that, under appropriate circumstances, choosing a basis that heavily pollute the standard projection method, significantly improves convergence rates of the quadratic projection method.

We have posted a fully functional Matlab code for assembling the matrices involved in the quadratic projection method forV a coulombic potential in the Manchester NLEVP collection of non-linear eigenvalue problems [BHM⁺08]. See the permanent link [BHM⁺]. In Appendix A we include details of the explicit calculation of all matrix coefficients.

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2. Spectral pollution and upper/lower spinor component balance

Consider, for any Ψ ∈ L²(R³)⁴, the spherical coordinates represen- tation:

(2.1) ψ(r, θ, φ) = rΨ(rsin(θ) sin(φ), rsin(θ) cos(φ), rcos(θ)) where (r, θ, φ) ∈ (0,∞) ×[0, π)× [−π, π). The map Ψ 7→ ψ is an isomorphism between L²(R³)⁴ and L²((0,∞), dr)⊗L²(S²)⁴. If we de- compose L²(S²)⁴ as the sum of the so called two-dimensional angular momentum subspaces K_m_j_,κ_j, the partial wave subspaces are given by

H_m_j_,κ_j =L²((0,∞), dr)⊗K_m_j_,κ_j, so thatL²(R³)⁴ =L

H_m_j_,κ_j. Without further mention, here and below we always assume that the indices (mj, κj) run over the set mj ∈ {−j,· · · , j} and κj ∈ {±(j +¹₂)}, for j ∈ {^2k+1₂ :k∈N}.

The r factor in (2.1) renders a Dirichlet boundary condition at 0.

The dense subspaces C₀^∞(0,∞)⊗K_m_j_,κ_j ⊂H_m_j_,κ_j are invariant under the action of H. If V is as in (1.1), then H ↾ C₀^∞(0,∞)⊗K_m_j_,κ_j is unitary equivalent to

(2.2) H_m_j_,κ_j :=

1 +φsc(r) +φel(r) −dr^d + ^κ_r^j +φam(r)

d

dr + ^κ_r^j +φam(r) −1−φsc(r) +φel(r)

. The operators Hmj,κj are essentially self-adjoint in C₀^∞(0,∞)² under suitable conditions on the potentials φsc,el,am. Then

Spec(H) =[

Spec(Hmj,κj).

Below we often suppress the sub-index (mj, κj) from operators and spaces, and only write the index κ≡ κj. Note that the eigenvalues of H are degenerate and their multiplicity is at least m_j. By virtue of (1.2),

Spec_disc(H) = [

Spec_disc(Hκ).

Since Spec_ess(Hκ) = (−∞,−1]∪[1,∞),Hκ are strongly indefinite.

Let us consider a heuristic approach to the problem of spectral pollution forHκ and the decomposition ofL²(0,∞)² into upper and lower spinor components. For simplicity we assume that the potential is purely electric and attractive, φsc(r) =φam(r) = 0 and φel(r)<0.

The pair (u, v)∈Dom(H_κ) is a wave function ofH_κ with associated eigenvalue E ∈(−1,1) if and only if

(2.3)

(φel+1−E)u+(−∂r+κ

r)v = 0 and (∂r+κ

r)u+(φel−1−E)v = 0.

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System (2.3) can be decoupled into

(2.4) LEu= 0 and v =−(φel−1−E)⁻¹

∂r+ κ r

u, where

L_E :=−

−∂_r+ κ r

(φ_el−1−E)⁻¹

∂_r+κ r

+ (φ_el+ 1−E)

≥(1 +E)⁻¹

∂r+κ r

_∗

∂r+κ r

+φel+ (1−E).

If we assumeφ_el is relatively compact with respect to the non-negative operator (∂_r+^κ_r)^∗(∂_r+ ^κ_r),

(2.5) min Spec_essLE ≥(1−E)>0.

Moreover, the expression for v in (2.4) yields u6≡ 0 and v 6≡ 0. Hence E ∈Spec_discHκ if and only if 0 is in the discrete spectrum of LE.

Let M^N ⊂ L²(0,∞) be a nested family of finite-dimensional subspaces such thatM^N ⊂ M^N+1,

[

N≥1

M^N =L²(0,∞)

and L^{N M} := M^N ⊕ M^M ⊂ Dom(Hκ). Let PN denote the orthogonal projection onto M^N, so that PN → I in the strong sense. With this notation we wish to apply the projection method to operatorHκ with test spaces L^{N M}.

We first consider the case N = M. Let (u_N, v_N)^t ∈ LN N be a sequence normalised by kuNk²+kvNk² = 1, such that

PN[(φel+ 1−EN)uN + (−∂r+κ

r)vN] = 0, (2.6)

P_N[(∂_r+ κ

r)u_N + (φ_el−1−E_N)v_N] = 0 (2.7)

for a suitable sequence EN →E˜ ∈(−1,1). Let χN be such that vN =−(φel−1−EN)⁻¹

∂r+κ r

uN +χN. By virtue of (2.7),

(2.8) P_N(φ_el−1−E_N)χ_N = 0.

If we where able to prove that (2.9) kPN(−∂r+ κ

r)χNk →0, N → ∞,

by substituting into (2.6), we would have kPNLE˜uNk → 0. Thus, by virtue of the min-max principle alongside with (2.5), we would have 0∈Spec_discLE˜ and so ˜E∈Spec_discHκ.

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Unfortunately, (2.9) can not be easily verified. Since PN → I in the strong sense, (φel −1 −EN)χN → 0 in the weak sense. Since φel < 0, χN → 0 also in the weak sense. Therefore we are certain that (−∂r + ^κ_r)χN → 0 and hence PNLE˜uN → 0 weakly. This does not imply in general (2.9), it only gives indication that the latter is a sensible guess.

The key idea behind the above heuristic argument motivates several pollution-free numerical methods for computing eigenvalues of H_κ, in- cluding the quadratic projection method discussed in the forthcoming section: as (2.3) might be vulnerable to spectral pollution in the gap (−1,1) because (2.9) is not necessarily guaranteed, we characterise the eigenvalues of this problem by testing whether 0 is in the spectrum of an auxiliary operator. This auxiliary operator has essential spectrum in {Re(z)>0}, so that a neighbourhood of 0 is always protected against spectral pollution (see the discussion preceding [Bou06, Lemma 5].) In the quadratic projection method, (2.9) is substituted by (3.4).

Remark 1. A successful procedure for computing eigenvalues of the Dirac operator is the one developed by Dolbeault, Esteban and S´er´e, [DES00] and [DES03]. This procedure systematically implements the above idea. Multiplying by u the left side equation of (2.4) and inte- grating in the space variable, gives A(E)u= 0 for

A(λ)v :=

Z _∞

0

|(r^κv)^′|²

r^2κ(1 +λ−φel)+ (φ_el+ 1−λ)|v|²dr.

Both terms inside the integral decrease in λ so, if v is regular enough, there is a unique λ=λ(v)∈R satisfying A(λ)v = 0. Upper estimates for the eigenvalues of Hκ are then found from those of the matrix corresponding to theλ-dependant formA(λ)v restricted tov ∈ Mⁿ.

Suppose now that M : N −→ N. Let (uN, vN)^t ∈ L^{N M(N)} be a sequence normalised bykuNk²+kvNk² = 1, such that (2.6) holds true and PN is replaced in (2.7) by PM(N). If limN→∞M(N)/N < 1, we are certainly less likely to obtain (2.9) as PN is replaced in (2.8) by PM(N). If, on the other hand, limN→∞M(N)/N > 1, then we would be more confident about obtaining (2.9) for symmetric reasons. In Section 4.4 we will present a series of numerical experiments supporting this heuristic argumentation. In particular, spectral pollution in the standard projection method appears to increase as limN→∞M(N)/N decreases (see figures 4) .

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3. The quadratic projection method

3.1. The second order relative spectrum. We now describe the basics of the quadratic projection method. In order to simplify the notation, below and elsewhereGdenotes a generic self-adjoint operator with domain Dom(G) acting on a Hilbert space. One should think of Gas being any of theHκ introduced in the previous section. The inner product in this Hilbert space is denoted byh·,·i and the norm byk · k. Let L ⊂ Dom(G) be a subspace of finite dimension. Assume that L = Span{b1, . . . , bn}, where the vectors bj are linearly independent.

Let

(3.1) K := (hGbj, Gbki)ⁿ_j,k=1, L:= (hGbj, bki)ⁿ_j,k=1 and B := (hbj, bki)ⁿ_j,k=1.

For z ∈ C, let Q(z) := Bz² −2zL +K ∈ Cⁿ×n. The aim of the quadratic projection method is to compute the so called second order spectrum of G relative to L:

Spec₂(G,L) := Spec(Q) ={λ∈C : Q(λ)v = 0, some 06=v ∈Cⁿ}. SinceB is a non-singular matrix and detQ(z) is a polynomial of degree 2n, Spec₂(G,L) consists of at most 2n points. These points do not lie on the real line, except if Lcontains eigenvectors of G. However, since Q(z)^∗ =Q(z),

Spec₂(G,L) = Spec₂(G,L).

Approximation of the discrete spectrum of G using the second order spectrum has been discussed in [LS04], [Bou07], [BL07] and the references therein.

The following result establishes a crucial connection between Spec(G) and Spec₂(G,L). Without further mention, we often identify the elements v ∈ L with correspondingv ∈Cⁿ in the obvious manner:

v =

n

X

k=1

hv, b^∗_jib_j and v = (hv, b^∗_ji)ⁿ_j=1,

where {b^∗_j} is the basis conjugate to {bj}. Note that if the bj are mutually orthogonal and kbjk= 1, then b^∗_j =bj. Below and elsewhere Π_S denotes the orthogonal projection onto a subspace S ⊂Dom(G).

Theorem 1. Let L ⊂Dom(G) be any finite-dimensional subspace. If λ∈Spec₂(G,L), then

(3.2) [Re(λ)− |Im(λ)|,Re(λ) +|Im(λ)|]∩Spec(G)6=∅.

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Moreover, suppose that E is an isolated eigenvalue ofGwith associated eigenspace E ⊂Dom(G). Let

dE := dist(E,SpecG\ {E}) = min{|E−x| : x∈SpecG, x6=E}. If

(3.3) [Re(λ)− |Im(λ)|,Re(λ) +|Im(λ)|]∩Spec(G) ={E} and Q(λ)v = 0 for 06=v ∈Cⁿ, then the corresponding v ∈ L satisfies

(3.4) kv−Π_Evk

kvk ≤ |Imλ| dE

.

Proof. Letλ∈Spec₂(G,L) and assume that Im(λ)6= 0. Since Q(z) = (h(z−G)bj,(z−G)bki)ⁿ_j,k=1,

Q(λ)v = 0 for non-trivialv ∈Cⁿ if and only if

(3.5) h(λ−G)v,(λ−G)wi= 0 ∀w∈ L. Foru, w∈ L and z =µ+iν where µ, ν ∈R, we have

h(z−G)u,(z−G)wi=h(µ−G)u,(µ−G)wi−ν²hu, wi−2iνh(µ−G)u, wi. In particular, if we take w=v in (3.5), we achieve

k(Reλ−G)vk²− |Imλ|²kvk²−2i|Imλ|h(Reλ−G)v, vi= 0.

Thus

(3.6) k(Reλ−G)vk

kvk =|Imλ| and

(3.7) |Imλ|h(Reλ−G)v, vi= 0.

But recall that G=G^∗, so

dist(x,SpecG) = min

u∈Dom(G)

k(x−G)uk kuk for x∈R. Therefore

dist(Reλ,SpecG)≤ |Imλ|, confirming (3.2).

For the second part, assume thatλ,EandE are as in the hypothesis, and let v satisfy (3.5) and hence (3.6). Then

k(E−G)vk ≤ |E−Reλ|kvk+k(Reλ−G)vk

≤2|Imλ|kvk.

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Since

dist(x,SpecG\ {E}) = min

u∈Dom(H), u⊥E

k(x−G)uk kuk , we have

k(E−G)(I−Π_E)vk ≥d_Ek(I−Π_E)vk, so that

k(I−Π_E)vk ≤ k(E−G)vk d_E . Now,

k(E−G)vk² =|E−Reλ|²kvk²+k(Reλ−G)vk²+2(E−Reλ)h(Reλ−G)v, vi. Thus, by (3.6) and (3.7),

k(E−G)vk² =|E−Reλ|²kvk²+|Imλ|²kvk² =|E−λ|²kvk². This gives

k(I−Π_E)vk ≤ |Imλ| dE kvk

as needed.

This theorem suggest a method for estimating Spec(G) from Spec₂(G,L). We call this method the quadratic projection method.

Choose a suitable L ⊂ Dom(G), find Q(z) and compute Spec₂(G,L).

Those λ ∈ Spec₂(G,L) which are close to R will necessarily be close to Spec(G), with a two-sided error given by |Im(λ)|. Moreover if λ is close enough to an isolated eigenvalue E of G, then

(3.8) |Reλ−E| ≤ |Imλ|

and a vector 06=v ∈ Lsuch thatQ(λ)v = 0 approaches the eigenspace associated to this eigenvalue with an error also determined by|Im(λ)|. Note that there is no concern with the position of E relative to the essential spectrum, or any semi-definitness condition imposed on G.

The procedure is always free from spectral pollution.

Remark 2. A stronger statement implying the first part of Theorem 1 can be found in [LS04, Lemma 5.2]. In fact, for isolated points of the spectrum, the residual on the right of (3.8) can actually be improved to

2|Im(λ)|²

dE for |Im(λ)| sufficiently small, see [BL07, Corollary 2.6]. How- ever, note that this later estimate is less robust in the sense that dE is not known a priori.

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3.2. The Hermite basis. In the forthcoming sections we apply the quadratic projection method to G = Hκ. To this end we construct finite-dimensional subspacesL ⊂Dom(Hκ) generated by Hermite functions.

Let the odd-order Hermite functions be defined by Φk(r) :=c⁻_2k+1¹ h2k+1(r)e⁻^r

2

2 , r≥0,

whereh_n(r) are the Hermite polynomials andc_n=p

2ⁿ⁻¹n!√πare nor- malisation constants. Motivated by the results of [Bou07, Section 3.4]

on Schr¨odinger operators with band gap essential spectrum, we choose (3.9)

L ≡ L^{N M} := Span

Φ₁(r) 0

, . . . ,

Φ_N(r) 0

,

0 Φ1(r)

, . . . ,

0 ΦM(r)

. Below we might consider an unbalance between the number of basis elements in the first and second component, N 6=M. Without further mention, we often write L^N ≡ L^{N N}, andLⁿ≡ L^N(n),M(n) when M and N depend upon n.

We now compute the matrix coefficients of Q(z). For this we recall some properties of Φ_k(r). The Hermite polynomials are defined by the identity

hn(z) := (−1)ⁿe^z² dⁿ

dzⁿe⁻^z², n ∈N. They satisfy the recursive formulae,

h^′_n(z) = 2nhn−1(z), (3.10)

hn+1(z) = 2zhn(z)−2nhn−1(z) (3.11)

and form an orthogonal set on the interval (−∞, ∞) with weight factor e⁻^z²,

Z _∞

−∞

h_m(z)h_n(z)e⁻^z²dz = 2ⁿn!√ πδ_nm. The generating function of this family of polynomials is

∞

X

n=0

hn(z)tⁿ

n! =e^2zt⁻^t². Thus

(3.12)

∞

X

n=0

hn(0)tⁿ n! =

∞

X

n=0

(−1)ⁿt²ⁿ

n! , so that hn(0) =

( 0 n−odd,

(−1)^n/2n!

(n/2)! n−even.

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The odd-order Hermite functions are the normalised wave functions of a harmonic oscillator,

(3.13) −Φ^′′_k(r) +r²Φk(r) = (2k+ 1)Φk(r),

subject to Dirichlet boundary condition at the origin. They form an orthonormal basis of L²(0,∞) and so B =I in (3.1).

The entries of the matrices K and L in (3.1) can also be found explicitly from known properties of the Hermite polynomials. The crucial terms for assembling these matrices are obtained in Appendix A.

3.3. Convergence. The procedure described above is useful, provided we can find points of Spec₂(H_κ,L) near the real axis. Here we formu- late sufficient conditions on the sequence of subspaces Lⁿ, in order to guarantee the existence of a sequence λn ∈ Spec₂(Hκ,Lⁿ) accumulating at points of the discrete spectrum of Hκ. We then show that the sequence of subspaces (3.9) satisfies these conditions.

We firstly recall the following result [Bou07, Theorem 2.1]. Note that L^{N M} are subspaces of Dom(H_κ²) for all κ.

Theorem 2. Let E be an isolated eigenvalue of finite multiplicity of G with associated eigenspace denoted by E. Suppose that Lⁿ⊂ Dom(G²) is a sequence of subspaces such that

(3.14) kG^p(u−Πnu)k ≤δ(n)kuk ∀u∈ E, p = 0,1,2,

where δ(n)→0asn → ∞is independent ofuandp. Then there exists b >0 and λn ∈Spec₂(G,Lⁿ), such that

(3.15) |λn−E|< bδ(n)^1/2.

We now verify (3.14) for G = Hκ and Lⁿ as in (3.9). To this end we consider an argument similar to the one discussed in [Bou07, Sec- tion 3.4] for the case of the cristaline non-relativistic Schr¨odinger operator.

Let

A=

A˜ 0 0 A˜

where ˜A=−∂_r²+r² acting onL²(0,∞), subject to Dirichlet boundary conditions at the origin. We fix the domain of A as

Dom(A) = [

N∈N

L^N,

so that A = A^∗, A has a compact resolvent and L^N ⊖ L^N−1 are the eigenspaces of A.

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Forf and g regular enough, we have Hκ

f g

=

p1f+p2g p3f+p4g

and H_κ² f

g

=

q1f +q2g q3f +q4g

, wherepj are linear polynomials andqj are quadratic polynomials in the variables∂r, φsc,el,am and κ/r. Condition (3.14) is achieved by showing that both Hκ and H_κ² are relatively bounded in the sense of operators with respect to A.

The following results can be easily extended to more general potentials. Here we only consider those of interest in our present discussion.

Lemma 3. Suppose that (3.16)

φsc,el,am=χsc,el,am+ψsc,el,am where

|χsc,el,am(r)| ≤cr⁻¹, ∀r >0, ψsc,el,am∈L^∞(0,∞),

for some constant c > 0. Then Dom(A) ⊆ Dom(H_κ²) and there exist constants a, b >0 such that

kH_κ^pvk ≤akAvk+bkvk ∀v ∈Dom(A), p= 0,1,2.

Proof. It is enough to check that the H_κ^p are relatively bounded with respect to −∂²_rIC². This, on the other hand, is a straightforward con-

sequence of Hardy’s inequality.

By combining this lemma with Theorem 2 and Theorem 1, we achieve the following.

Corollary 4. LetE be an isolated eigenvalue of Hκ of finite multiplicity with associated eigenspaceE. Let L^{N M} be defined by (3.9). Suppose that φsc,el,am satisfy (3.16) and assume additionally that E ⊂ Dom(A^q) for some q >1. For b >0 large enough and independent of N or M, we can always find a sequence λN M ∈Spec₂(Hκ,L^{N M}), such that (3.17) |λ_{N M}−E|< b(N⁻^q−1² +M⁻^q−1² ) and

k(I−Π_E)vN Mk< b(N⁻^q⁻¹² +M⁻^q⁻¹² ), where vN M ∈ L^{M N} solves Q(λN M)vN M = 0.

Proof. Let v = v1

v₂

∈ E normalised by kvk = 1. By virtue of Lemma 3 and Theorem 2, the desired conclusion follows if we are able to find b >0 such that

(3.18)

X∞

k=n

|hA˜^rv_j,Φ_ki|² < bn⁻^2(q⁻^r),

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for r = 0,1 and j = 1,2. In order to show (3.18), note that the hypothesis v ∈Dom(A^q) ensures

(2n)^2(q⁻^r)

∞

X

k=n

|hA˜^rvj,Φki|² ≤

∞

X

k=n

(2k+ 1)^2q|hvj,Φki|²

=

∞

X

k=n

|hA˜^qvj,Φki|² →0

asn → ∞.

Remark 3. Suppose that the number of degrees of freedom M+N = 2nis fixed. Modulo the constantb, the bound on the right side of (3.17) is optimal at (N, M) = (n, n). This suggests that an optimal rate of approximation might be achieved by choosing an equal number of upper/lower spinor components in (3.9). Contrary to this presumption, and depending on the potential V, the numerical evidence we present in sections 4.4 and 4.5, show that the residual on the right side of (3.8) can in some cases decrease significantly (over 18% in some cases) by suitably choosing N 6=M.

We now explore precise conditions on the potential, in order to guarantee the hypothesis of Corollary 4.

Lemma 5. Let φsc,el,am ∈ C^∞(0,∞) be such that φsc,el,am(r) → 0 as r→ ∞. Assume additionally thatr7→r^αφsc,el,am(r)are locally bounded for some α∈(0,1). Let Hκu =Eu. For sufficiently small a >0,

ke^aruk^H^p^(0,∞)<∞ ∀p∈N. Proof. See [BG87, Corollary 3.1]. Let

D_κ =

1 −dr^d + ^κ_r^j

d

dr +^κ_r^j −1

and V =

φsc+φel φam

φam −φsc+φel

. For any 0 < ε < min|E ±1|, we can always separate V = Vc +Vε

where:

(a) Vc is smooth, it has compact support and a singularity of order O(r⁻^α) at the origin,

(b) Vε is smooth with bounded derivatives and kVεk∞< ε.

Then

u=−(Dκ+Vε−E)⁻¹Vcu.

Multiplying this identity by D^p_κ and e^ar, yields

(3.19) D_κ^pe^aru=−D_κ^p(Dκ+Vε+ia−E)⁻¹e^arVcu.

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Let σ ∈ R. On the one hand, (Dκ +Vε +ia−E)⁻¹ is a bounded operator from H^σ(0,∞) to H^σ+1(0,∞). On the other hand, multipli- cation by Vc is a bounded operator from H^σ(0,∞) into H^σ⁻^α(0,∞).

Indeed, note that [Tri99, Theorem 2.1(i) forp=q= 2] yield the latter result forσ ∈N after commutation and iteration, then duality and in- terpolation ensure it for allσ∈R. Therefore, as (Dκ+Vε+ ia−E)⁻¹Vc

is bounded from H^σ(0,∞) to H^σ+(1⁻^α)(0,∞), (3.19) and a standard bootstrap argument ensure the desired conclusion.

We remark that, by using [His00, Lemma 5.1] (which can be modified in order to get ride of the term ¹₂ in its assumption (i)), necessarily a <p

(1−ε)²−E² for the above lemma to hold true.

Corollary 6. Assume thatφsc,el,am are as in Lemma 5. LetE be an isolated eigenvalue of Hκ of finite multiplicity with associated eigenspace E. Let L^{N M} be defined by (3.9). For b >0 large enough and independent ofN orM, we can always find a sequenceλN M ∈Spec₂(Hκ,L^{N M}), such that (3.17) holds true ifq <5/4.

Proof. Letv ∈ E be as in the proof of Corollary 4. We show that (3.20)

∞

X

k=0

(2k+ 1)^2q|hvj,Φki|² <∞,

for j = 1,2. LetF = (∂r−r). Integration by parts and (3.10), yield hvj,Φki= 1

c2k+1

Z _∞

0

vj(r)h2k+1(r)e⁻^r²2 dr

= 1

2(2k+ 2)c2k+1

Z _∞

0

vj(r)h^′_2k+2(r)e⁻^r

2 2 dr

= −1

2(2k+ 2)c_2k+1 Z _∞

0

F vj(r)h2k+2(r)e⁻^r

2 2 dr

= 1

2²(2k+ 2)(2k+ 3)c2k+1

Z _∞

0

F vj(r)h^′_2k+3(r)e⁻^r

2 2 dr

= 1

2²(2k+ 2)(2k+ 3)c2k+1

Z _∞

0

F²vj(r)h2k+3(r)e⁻^r

2 2 dr

= −1

2³(2k+ 2)(2k+ 3)(2k+ 4)c2k+1

F²v_j(0)h_2k+4(0) + Z _∞

0

F³v_j(r)h_2k+4(r)e⁻^r

2 2 dr

= −F²vj(0)h2k+4(0)

2³(2k+ 2)(2k+ 3)(2k+ 4)c2k+1

+ R_∞

0 F⁴vj(r)h2k+5(r)e⁻^r

2 2 dr 2⁴(2k+ 2). . .(2k+ 5)c2k+1

=a₁+a₂. Identity (3.12) alongside with the fact that|F²v_j(0)|<∞(see Lemma 5) and the Stirling formula, ensures that a1 ∼ n⁻^7/4 as n → ∞. On the

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other hand, Lemma 5 ensures that F⁴vj ∈L²(0,∞). Thus, since c2k+5

2⁴(2k+ 2). . .(2k+ 5)c2k+1 ∼n⁻²,

a2 =O(n⁻²) asn → ∞. This guarantees (3.20) forq <5/4.

According to this corollary, for smooth potentials, the order of approximation of the quadratic projection method to any eigenvalue E of Hκ should be at least a power 1/8 of the dimension of L^{M N}. The numerical experiment performed in Section 3.3 show that this bound improves substantially for particular potentials. See Figure 8 (right) and Table 5 (right).

Remark 4. The arguments involved in the proof of this theorem show that, the behaviour of the wave functions at the singularity and its regularity are the main ingredients responsible for controlling the speed of approximation when using a Hermite basis (3.9).

4. Some numerical experiments

We now report on various numerical experiments performed for very simple radially symmetric potentials. It is not our intention to show accurate computations, but rather to illustrate how the method discussed in Section 3 can be implemented in order to rigourously enclose eigenvalues and compute eigenfunctions of the Dirac operator.

4.1. Ground state of the purely coulombic potential. We begin by considering the analytically solvable case ofV being a radially symmetric purely coulombic potential: φsc =φam = 0, φel(r) =γ/r. Here

−√

3/2< γ < 0. The eigenvalues ofHκ are given explicitly by

(4.1) Ej = 1 + γ²

(j +p

κ² −γ²)²

!₋1/2

.

Note thatEj →1 asj → ∞for all values ofκ. The ground state of the full coulombic Dirac operatorH is achieved when κ =−1 and j = 0.

In Figure 1 we superimpose the computation of Spec₂(H₋1,Lⁿ) for two values of n, in a narrow box near the interval [−3,3]. Here Lⁿ is given by (3.9). For the set of parameters considered (κ = −1 and γ = −1/2), (4.1) yields E0 ≈ 0.866025, E1 ≈ 0.965925, E2 ≈ 0.9851210, E3 ≈ 0.99174012 and E4 ≈ 0.9947623. A two sided approximation of E0 is achieved from the point λ∈Spec₂(H₋1,L¹⁰⁰⁰) at λ≈0.8661+0.0236i. According to (3.2), there should be an eigenvalue ofH₋1 in the interval [0.8661−0.0236,0.8661+0.0236]. This eigenvalue happens to beE0. ForE1,E2 and the pair (E3,E4), we can also derive

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similar conclusions. Note that Spec_ess(H₋1) is also revealed by points of Spec₂(H₋1,Lⁿ) seemingly accumulating at (−∞,−1]∪[1,∞).

In Figure 2 we show approximation of the corresponding ground wave function associated withE0. We have also depicted the analytical eigenfunction:

(4.2) u0(r) = ν0

γ

(1−γ²)^1/2−1

r√

1−γ²e⁻^(γE⁰^/√

1−γ²)r,

where ν0 is chosen so that ku0k= 1. From this picture it is clear that, at least qualitatively, u0(r) seems to be captured quite well even for small values of n.

We show a quantitative analysis of the calculation of u0 in Table 3.

In the middle column we compute the residual on the left of (3.4) and on the last column we compute the right hand side of (3.4). It is quite remarkable that the actual residuals are over 74% smaller than the error predicted by Theorem 1.

4.2. β-Dependence of the sub-coulombic potential. We now investigate the case of the potential being radially symmetric and sub- coulombic: φsc = φam = 0, φel(r) = γ/r^β for β ∈ (0,1). Here

−1 < γ < 0. The purpose of this experiment is to show how The- orem 1 provides a priori information about Spec_disc(Hκ) even for small values of n. Note that (1.2) is guaranteed from [Tha92, Theorems 4.7 and 4.17]. FurthermoreH has infinitely many eigenvalues according to [Tha92, Theorem 4.23].

In Figure 3, we show computation of the ground state of H₋₁, for β = 0.1 : 0.1 : 1 and γ =−1/2. As β → 1⁻, E₀ →0.8661, the ground eigenvalue of the coulombic Dirac operator. Asβ →0⁺ the eigenvalue remains above 1/2. Note that the family of operatorsHκis not analytic at β= 0 for this potential. For β = 0 the spectrum becomes

Spec(Hκ) = (−∞,−3/2]∪[1/2,∞).

The vertical bars show|Im(λ)|, the maximum error in the computation of E₀ ≈ Re(λ) given by Theorem 1. For this example we have chosen n = 15. Table 4 contains the data depicted in Figure 2. Observe that the error increases as β → 0⁺ and β → 1⁻. This seems to be a consequence of the fact thatE₀ becomes closer to other spectral points at both limits, so dE0 →0.

4.3. The inverse harmonic electric potential. In this set of experiments we consider another canonical example in the theory of Dirac operators: φsc =φam = 0, φel(r) = γ/(1 +r²) for γ < 0. The discrete spectrum of H is known to be finite for −1/8 < γ < 0 and infinite

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for γ < −1/8, [Kla80]. As the parameter γ decreases, we expect that eigenvalues will appear at the threshold 1, move through the gap, and leave it at -1. This dynamics is shown in Figure 4, for the ground eigenvalue of H₋1. It is a long standing question whether the eigenvalues become resonances when they re-enter the spectrum.

In Figure 5 we depict the first three eigenfunctions of H₋1 for γ =

−4. They correspond to the eigenvalues E0 ≈ −0.3955, E1 ≈ 0.6049 and E2 ≈ 0.9328. See also Figure 7. Note that for γ = −4, both components of the eigenfunctions appear to obey a Sturm-Liouville type oscillation hierarchy.

4.4. Upper/lower spinor component balance and approximation of eigenvalues. We now investigate the effects of “unbalancing”

the basis by choosing N 6=M.

In Figure 6, we have performed the following experiment. Fix the number of degrees of freedom, dim(L^{M N}) = 200. Then for N = 10 : 5 : 190 and M = 200−N, use the quadratic method as well as the Galerkin method to approximate eigenvalues ofH_κ in the spectral gap (−1,1). We firstly consider φsc=φam = 0 and φel(r) = −1/(2r).

The Galerkin method might or might not produce spurious eigenvalues. The quadratic method will always provide two-sided non-polluted bounds for the true eigenvalues with a residual, obtained from (3.2), which might change with N. See also figures 4 and 6. The Galerkin method appears to pollute heavily near the upper end of the gap for N > M, as predicted by the considerations of Section 2. Moreover, for the ground state, the minimal |Im(λ)| is not achieved at N = 100 which corresponds to N = M, but rather at some N > 100. It is remarkable that the residual are reduced significantly (up to 66% for the true residual) when M(N)/N ≈1/5.

If we performed the analogous experiment for the inverse harmonic potential, the conclusion are also rather surprising. See Figure 7. The Galerkin method appears to pollute heavily near the upper end of the gap for N > M as predicted in Section 2. However, now the approximation is improved by over 16% for E0 and over 18% for E1, if M(N)/N ≈3.

We can explain these phenomena by considering the relation between the components of the exact eigenvectors.

In the case of a purely coulombic potential, the ground state is given by (4.2) where ν0 is a real constant. The lower spinor component just differs from the upper one by a scalar factor. When γ ∈ (0,1), the lower component is smaller in modulus than the upper one. Choosing N > M, can reduce an upper bound of the residual associated to the

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first component, while the residual associated to the second component remains small due to the smallness of the lower component.

In the case of an inverse harmonic purely electric potential, this argument fails, as the two spinor components of the eigenfunction are not a scalar factor of each other, see Figure 5. If we denote an eigenfunction by u= (u^upp, u^low), the figure suggests that|∂_r²u^upp(0)|<< |∂_r²u^low(0)|. As|∂²_rΦk(0)|= 0, it is natural to expect that a decrease in the residual is only achieved by choosing a suitable M > N.

Remark 5. Although we can not prove it rigourously, strong evidence suggests that (for any of the potentials considered above) no spurious eigenvalue is produced by the Galerkin method when N = M. Why bothering then with more complicated procedures, such as the quadratic projection method, to avoid inexistent spectral pollution. A partial answer is, on the one hand, robustness: we do not know a priori whether the Galerkin method pollutes for a given basis. On the other hand, as the experiments of this section suggest, some times forcing a kinetic unbalance into a model might improve convergence properties.

4.5. Convergence properties of the odd Hermite basis. A convergence analysis, as the number of degrees of freedom increases, can be found in Figure 8 and Table 5. Due to the discussion of Section 4.4, we consider this for different ratios N/M.

The right graph shows that the conclusion of Corollary 6 is far from optimal for the inverse harmonic potential of Section 4.3. As expected from Section 4.4, a faster convergence rate as well as smaller residuals are found if we suitably choose N < M.

The left graph corresponds to the coulombic potential discussed in Section 4.1. It clearly shows that the order of convergence of λn →E does not obey the estimate |λ_n− E| ≤ O(n⁻^a) (for some a > 0) of Corollary 4. In fact the convergence rate seems to decrease as we increase the number of degrees of freedom. This reduction in the speed of convergence can be prevented by puttingM =f(N) for a suitable non- linear increasing function 0 < f(x) < x. The optimal f(x), however, might depend on the eigenvalue to be approximated.

Remark 6. According to Remark 2, the actual approximate eigenvalue Re(λ) is correct up toO(n⁻^2a), whereais the second column of Table 5.

Furthermore, note that, in the case of the coulombic potential we can compute directly the true residual|Re(λ)−E|. From Figure 6 bottom, it is clear that this true residual is substantially smaller than the one estimated by|Im(λ)|.

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Appendix A. Entries of the matrix polynomial coefficients

The recursive identities satisfied by the Hermite functions allow us to find recursive expressions for the matrix entries of K and L in (3.1) when G =Hκ. Rather than estimating the corresponding inner prod- ucts by trapezoidal rules, we build the codes involved in the numerical experiments performed in Section 4 using these explicit expressions. As large factors are cancelled in these explicit expressions, this approach turns out to be far more accurate. Since some of the calculations are not entirely trivial, we include here the crucial details.

Let T1 =

Z _∞

0

Φk(r)Φj(r) dr, T2(k, j) = Z _∞

0

Φ^′_k(r)Φj(r) dr, T3 =

Z _∞

0

1

rΦk(r)Φj(r) dr, T4 = Z _∞

0

Φ^′_k(r)Φ^′_j(r) dr, T5(k, j) =

Z _∞

0

1

rΦ^′_k(r)Φj(r) dr, T6 = Z _∞

0

1

r²Φk(r)Φj(r) dr, F1 =

Z _∞

0

φel(r)Φk(r)Φj(r) dr, F2 = Z _∞

0

φ²_el(r)Φk(r)Φj(r) dr, F3 =

Z _∞

0

φel(r)

r Φk(r)Φj(r) dr, F4(k, j) = Z _∞

0

φel(r)Φ^′_k(r)Φj(r) dr.

Here and below we stress the dependence on j, k when the coefficient is not symmetric with respect to these indices. Denote

Ψk,1 = Φ_k

0

, Ψj,2 = 0

Φj

.

ThenhHκΨkl,Ψjmiare given according to Table 1 andhHκΨkl, HκΨjmi are given according to Table 2.

m= 1 m= 2

l = 1 T1 +F1 T2(k, j) +κT3

l = 2 −T2(k, j) +κT3 −T1 +F1

Table 1. Term hHκΨkl,Ψjmi.

Form, n∈N∪0, let P(n) =

Qn

l=1 1 + _2l¹

n 6= 0

1 n = 0

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m= 1 m = 2

l = 1 T₁+T₄+κT₅(k, j) +κT₅(j, k)+ −T₂(k, j)−T₂(j, k) + 2κF₃+ κ²T6+ 2F1+F2 F4(k, j)−F4(j, k)

l = 2 −T2(k, j)−T2(j, k) + 2κF3+ T1+T4−κT5(k, j)−κT5(j, k)+

−F4(k, j) +F4(j, k) κ²T6−2F1+F2

Table 2. Term hHκΨkl, HκΨjmi.

and

(A.1) I(m, n) = 1

c_mc_n Z _∞

0

hm(r)hn(r)e⁻^r²dr.

Lemma 7.

I(m, n) =







δmn m≡n (mod 2)

(−1)^k⁻^j+1√

2P(k)P(j) (2k−2j−1)√

π(2k+1) m= 2k, n= 2j + 1.

Proof. Ifm≡n (mod 2), thenhm(r)hn(r) is an even function forr∈R and so

Z _∞

0

hm(r)hn(r)e⁻^r²dr = 1 2

Z _∞

−∞

hm(r)hn(r)e⁻^r²dr.

On the other hand, if m 6≡ n (mod 2), say m = 2k and n = 2j + 1, (3.10) and integration by parts yield

Z _∞

0

h2k(r)h2j+1(r)e⁻^r²dr= Z _∞

0

h2j+1(r)(e⁻^r²)^(2k)dr

=− Z _∞

0

h^′_2j+1(r)(e⁻^r²)^(2k⁻¹⁾dr

=−2(2j+ 1) Z _∞

0

h_2j(r)(e⁻^r²)^(2k⁻¹⁾dr

= 2²(j + 1)(2k−1) Z _∞

0

h2k−2(r)h2j−1(r)e⁻^r²dr.

The corresponding expression forI(m, n) can be obtained in a straight-

forward manner from these two assertions.

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Lemma 8.

T₁(k, j) =δ_jk, T₂(k, j) = 4(−1)^k⁻^j+1(k−j)

√π(2k−2j −1)(2k−2j + 1)

pP(j)P(k), T3(k, j) = 2(−1)^k⁻^j+1p

P(j)

√πp P(k)

k

X

m=0

P(m)

(2m+ 1)(2m−2j−1),

T4(k, j) = 1 2











−p

2k(2k+ 1) j =k−1

4k+ 3 j =k

−p

(2k+ 2)(2k+ 3) j =k+ 1

0 otherwise,

T5(k, j) =







2(−1)^j⁻^kq

P(k)

P(j) k < j

1 k=j

0 k > j,

T6(k, j) = (−1)^j⁻^k2







qP(j)

P(k) j ≤k qP(k)

P(j) k < j.

Proof. LetI(m, n) be given by (A.1). By virtue of identities (3.10) and (3.11),

Z _∞

0

h^′_2k+1h2j+1e⁻^r²dr = (2k+ 1)I(2k,2j+ 1)−2⁻¹I(2k+ 2,2j+ 1), Z _∞

0

1

rh2k+1h2j+1e⁻^r²dr=

k

X

l=0

(−1)^l2^2l+1 k!

(k−l)!I(2(k−l),2j+ 1).

This yields T₂ and T₃. Let

J(k, j) = Z _∞

0

1

rh2kh2j+1e⁻^r²dr.

Then

J(k, j) =

^√_π2²j(2k)!(−1)^j⁻^kj!

k! k≤j

0 k > j

and Z _∞

0

1

rh^′_2k+1h2j+1e⁻^r²dr = 2(2k+ 1)J(k, j)−δkj√

π2^k(2k+ 1)!.

This renders T5. Moreover, integration by parts ensures T6 =

Z _∞

0

1

r²ΦkΦj =− Z _∞

0

1 r

_′

ΦkΦj = T5(k, j) +T5(j, k) . The expression for T4 follows from (3.13) and the identity

Z _∞

0

Φ^′_k(r)Φ^′_j(r) dr = 1 2

Z _∞

−∞−Φ^′′_k(r)Φj(r) dr.

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From E1 and E2 in the next lemma, one easily obtains explicit formulae for Fn when φel(r) =γ/r^β.

Lemma 9. Forβ ∈[0,1] and α∈[−1,2], let E1(β, k, j) =

Z _∞

0

1

r^βΦ^′_k(r)Φj(r) dr, E2(α, k, j) = Z _∞

0

1

r^αΦk(r)Φj(r) dr.

Then

E1(β, k, j) = (2j+1)E2(β+1, k, j)+p

2(2j+ 1)jE2(β+1, k, j−1)−E2(β−1, k, j) and

E₂(α, k, j) = 2P(j)P(k)(−1)^k+j

√π

k,j

X

m,p=1

(−1)^m+pΓ(³⁻₂^α +m+p) k

m j p

m!p!P(m)P(p) . Proof. Let

Sn(k) =c⁻_2k+1¹ (2k+ 1)! (−1)^k⁻ⁿ2²ⁿ⁺¹ (k−n)!(2n+ 1)!. Then

c⁻_2k+1¹ h2k+1(r) =

k

X

n=0

Sn(k)r²ⁿ⁺¹

and

E2(α, k, j) =

k,j

X

m,p=1

Sm(k)Sp(j)K(α, m, p)

where

K(α, m, p) = Z _∞

0

1

r^αr^2m+1r^2p+1e⁻^r²dr = 1

2Γ3−α

2 +m+p . On the other hand, the expression for E1 follows from applying (3.10)

and (3.11).

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If E3 and E4 are as in the following lemma andφel(r) = _1+r¹2, then F₁(k, j) = E₃(2k+ 1,2j+ 1),

F2(k, j) = 1 2

r2k+ 1

2 E4(2k,2j + 1) +

r2j+ 1

2 E4(2k+ 1,2j)

−√

k+ 1E4(2k+ 2,2j+ 1)−p

j+ 1E4(2k+ 1,2j+ 2)+

E3(2k+ 1,2j+ 1) , F3(k, j) = T3(k, j)−

r2k+ 1

2 E3(2k,2j+ 1)−√

k+ 1E3(2k+ 2,2j+ 1), F4(k, j) =

r2k+ 1

2 E3(2k,2j+ 1)−√

k+ 1E3(2k+ 2,2j+ 1).

Lemma 10. Form, n∈N∪ {0}, let I(m, n) be as in Lemma 7, E3(m, n) = 1

cmcn

Z _∞

0

1

1 +r²hm(r)hn(r)e⁻^r²dr, E4(m, n) = 1

cmcn

Z _∞

0

r

1 +r²hm(r)hn(r)e⁻^r²dr.

Then

E3(0,0) = 2e Z 1

0

e⁻^t²dt+e√ π, E4(0,0) =e√

π Z _∞

1

e⁻^t t dt, E3(m+ 1,0) = 1

√m+ 1 E4(m,0)−√

mE3(m−1,0) , E4(m+ 1,0) = 1

√m+ 1 √

2I(m,0)−√

2E3(m,0)−√

mE4(m−1,0) , E3(m+ 1, n+ 1) = 1

p(m+ 1)(n+ 1) 2I(m, n)−2E3(m, n)+

−√

2mE4(m−1, n)−√

2nE4(m, n−1) +√

mnE3(m−1, n−1) E₄(m+ 1, n+ 1) = 1

p(m+ 1)(n+ 1)

p2(n+ 1)I(m, n+ 1)−2E₄(m, n)+

√2nE3(m, n−1)−√

2mE4(m−1, n) +√

mnE3(m−1, n−1) . Proof. The recursions for E3 and E4, follow from (3.11).